Question: Two cards are chosen at random from a standard 52-card deck.  What is the probability that the first card is a heart and the second card is a 10?
Solution: There are two cases.

Case 1: The first card is a $\heartsuit$ but not a 10.

The probability of the first card satisfying this is $\dfrac{12}{52},$ and then the probability that the second card is a 10 is $\dfrac{4}{51}.$

Case 2: The first card is the 10 $\heartsuit$.

The probability of the first card being the 10 $\heartsuit$ is $\dfrac{1}{52},$ and then the probability that the second card is a 10 is $\dfrac{3}{51}.$

We then add the probability of the two cases (since they are exclusive) to get \[\frac{12}{52}\times \frac{4}{51}+\frac{1}{52}\times \frac{3}{51}=\boxed{\frac{1}{52}}.\]